The parity conjecture for 2-torsion in the Tate–Shafarevich group

Let $K$ be a number field and let $E$ be an elliptic curve defined over $K$. The group of $2$-torsion points in the Tate–Shafarevich group is $\Sha(E/K)[2]$, viewed as an $\mathbb{F}_2$-vector space. Parity conjecture. For every elliptic curve $E$ defined over $K$,

$$\dim_{\mathbb{F}_2} \Sha(E/K)[2]$$

is even. This is a well-known conjecture that follows from the Tate–Shafarevich conjecture, and it is used to obtain results about quadratic twists of elliptic curves having Mordell–Weil rank one.